TRAVELING-WAVE TUBES 17 



at about the same speed. In the backward direction there is no cumulative 

 action, because the wave and the electrons are moving in the opposite 

 directions. 



The variation in the z direction for three forward waves is as 



exp —Tz = exp —jjSeZ exp dC^^^ (2.33) 



We see that the first wave is an increasing wave which travels a little more 

 slowly than the electrons. The second wave is a decreasing wave which 

 travels a little more slowly than the electrons. The third wave is an un- 

 attenuated wave which travels faster than the electrons. It can be shown 

 generally that when a stream of electrons interacts with a wave, the electrons 

 must go faster than the wave in order to give energy to it. 



It is interesting to know the ratio of line voltage to line current, or the 

 characteristic impedance, for the three forward waves. This may be obtained 

 from (2.5). We see that the characteristic impedance Kn for the nth. wave is 

 given in terms for the propagation constant for the nth. wave, r„, by 



Kn = V/I = yX/r„ (2.34) 



In terms of 5„ this becomes 



A',. = K{J - l3eC8n/ry) (2.35) 



Kn = K{1 - jC8n) (2.36) 



We see that the characteristic impedance for the forward waves differs from 

 the characteristic impedance in the absence of electrons by a small amount 

 proportional to C, and that the characteristic impedance has a small reactive 

 component. 



We are particularly interested in the rate at which the increasing wave 

 increases. In a number of wave lengths N, the total increase in db is given by 



20 logio exp [(\/3/2)(C)(27riV)] db 



= 47.3 CN db ^^-^^^ 



We will see later that the overall gain of the traveling-wave tube with a 

 uniform helix can be expressed in the form 



G = A -\- BCN db (2.38) 



Here yl is a loss relating voltage associated with the increasing wave to 

 the total applied voltage. This loss may be evaluated and will be evaluated 

 later by a proper examination of the boundary conditions at the input of 

 the tube. It turns out that for the case we have considered 



G = -9.54 + 47.3 CN db (2.39) 





